How Rational Behavior Leads to Inefficiency if there are Incomplete Markets

Robert Waldmann

My effort immediately below to explain some general equilibrium theory in English didn’t work out so well. Here I will attempt to give simple examples which show how rational individual choice and/or trade between rational consenting adults can make everyone worse off.

Models will all involve strange fruit trees, that is assets which generate goods without labor. There will be only 2 periods. In the first period (period 0) agents plant trees and maybe buy and sell them. It is allowed to short an apple tree (that is to promise to deliver the apples even if you don’t own the tree).

Then in period 1 the trees produce fruit. The amount depends on the state of the world (that is say the weather). People fulfill the terms of their financial agreements. Then people trade fruit on the spot market. Then people eat the fruit giving them their only pleasure in the whole model. Then they die.

If there are so many different kinds of trees that, for each state of the world, one can construct a portfolio which pays a positive amount in that state and only in that state, then markets are complete and the free market outcome is Pareto efficient. Otherwise it is easy to come up with examples in which rational behavior in period 0 makes everyone worse off in expected value.

One Example after the jump (I will add more when my fingers stop aching).

drought resistent apple trees. Here there are only 2 states : in state 1 it rains a lot on the apple orchards, in state 2 it rains little. The only choice in period zero is that people with apple suitable land have to decide wether to grow drought resistant apple trees which get mold if it is wet or drought sensitive apple trees which do great if it is wet. People who grow apples get sick of them and don’t want to eat them. They just eat the other good (oranges). There are always the same number of oranges. Orange growers have utility which is cobb douglas in apples and oranges.

Now if all apple growers plant drought resistent apples they all face no risk. If there are half as many apples, the relative price of apples doubles. They always eat the same number of oranges (and no apples they don’t like apples). If one apple grower plants any drought sensitive apples and their is a drought, he will suffer. If apple growers are risk averse enough, all drought resistant apples will be an equilibrium.

The orange growers face risk. They always eat the same number of oranges but eat different numbers of apples depending on whether there is a drought.

A Pareto improvement is possible. If all the apple growers plant half and half drought resistent and drought sensitive apples, then they are still safe and so are the orange growers. This is a Pareto improvement.

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